Functional convergence of stochastic integrals with application to statistical inference

Assuming that { ( U n , V n ) } is a sequence of càdlàg processes converging in distribution to ( U , V ) in the Skorohod topology, conditions are given under which { ∬ f n ( β , u , v ) d U n d V n } converges weakly to ∬ f ( β , x , y ) d U d V in the space C ( R ) , where f n ( β , u , v ) is a sequence of “smooth” functions converging to f ( β , u , v ) . Integrals of this form arise as the objective function for inference about a parameter β in a stochastic model. Convergence of these integrals play a key role in describing the asymptotics of the estimator of β which optimizes the objective function. We illustrate this with a moving average process.